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          <h1 class="post-title" itemprop="name headline">What is an RKHS [2]</h1>
        

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        <p>第二部分介绍RKHS的概念，以及核函数的构造。<br><a id="more"></a></p>
<h2 id="Reproducing-kernel-Hilbert-space"><a href="#Reproducing-kernel-Hilbert-space" class="headerlink" title="Reproducing kernel Hilbert space"></a>Reproducing kernel Hilbert space</h2><h4 id="Definition-of-an-RKHS"><a href="#Definition-of-an-RKHS" class="headerlink" title="Definition of an RKHS"></a>Definition of an RKHS</h4><h4 id="Evaluation-functional"><a href="#Evaluation-functional" class="headerlink" title="Evaluation functional"></a>Evaluation functional</h4><p>假设一个元素是函数 $f:\mathcal{X}\rightarrow\mathbb{R}$ 的定义在$\mathcal{X}$ 的 Hilbert Space $\mathcal{H}$，也写作（$\mathcal{H}\subset\mathbb{R}^\mathcal{X}$）。对于固定的$x\in\mathcal{X}$，映射$\delta_x:\mathcal{H}\rightarrow\mathbb{R}$$\delta_x:f\mapsto f(x)$称为Dirac evaluation functional</p>
<p>显然Evaluation functional是线性的，但是无法确定连续（等价于有界）。</p>
<h4 id="Reproducing-kernel-Hilbert-space-1"><a href="#Reproducing-kernel-Hilbert-space-1" class="headerlink" title="Reproducing kernel Hilbert space"></a>Reproducing kernel Hilbert space</h4><p>如果$f:\mathcal{X}\rightarrow\mathbb{R}$的 Hilbert Space $\mathcal{H}$，其$\delta_x$连续,$\forall x\in\mathcal{X}$。那么$\mathcal{H}$为RKHS</p>
<p>在RKHS中，范数收敛可以得到点收敛。如果两个函数在RKHS范数中收敛，那么他们在任意一个点收敛。即</p>
<script type="math/tex; mode=display">\text{if} \lim_{n\rightarrow\infty}||f_n-f||_\mathcal{H}=0\text{ then }\lim_{n\rightarrow\infty}f_n(x)=f(x),\forall x\in\mathcal{X}</script><p>反之，如果不再RKHS中，则无法从范数收敛推出任一点收敛。</p>
<h4 id="Reproducing-kernels"><a href="#Reproducing-kernels" class="headerlink" title="Reproducing kernels"></a>Reproducing kernels</h4><p>假设$\mathbb{R}$-valued 函数的Hilbert space $\mathcal{H}$，函数$k : \mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$是$\mathcal{H}$的再生核函数当其满足</p>
<ul>
<li>$\forall x\in\mathcal{X},k(\cdot,x)\in\mathcal{H}$</li>
<li>$\forall x\in\mathcal{X},\forall f\in\mathcal{H},\langle f,k(\cdot,x)\rangle_\mathcal{H}=f(x)$(reproducing)</li>
</ul>
<p>可以得出</p>
<script type="math/tex; mode=display">k(x,y)=\langle k(\cdot,x),k(\cdot,y)\rangle_\mathcal{H}</script><p>当$\mathcal{H}$的再生核函数存在即唯一。$\mathcal{H}$是RKHS$\Leftrightarrow$$\mathcal{H}$有再生核函数（可用里斯表示定理证明）</p>
<h4 id="Feature-space-and-other-kernel-properties"><a href="#Feature-space-and-other-kernel-properties" class="headerlink" title="Feature space, and other kernel properties"></a>Feature space, and other kernel properties</h4><h4 id="Kernels"><a href="#Kernels" class="headerlink" title="Kernels"></a>Kernels</h4><p>$\mathcal{X}$为非空集合，函数$k : \mathcal{X}\times\mathcal{X}\rightarrow\mathbb{R}$是核函数，当且仅当存在一个Hilbert space $\mathcal{H}$ 和映射$\phi:\mathcal{X}\rightarrow\mathcal{H}$ s.t.</p>
<script type="math/tex; mode=display">k(x,y)=\langle\phi(x),\phi(y)\rangle_\mathcal{H}</script><p>映射$\phi:\mathcal{X}\rightarrow\mathcal{H}$称作特征映射，$\mathcal{H}$称作特征空间。对于给定的核函数，可能有多个特征映射。</p>
<h2 id="Construction-of-an-RKHS-from-a-kernel-Moore-Aronsaj"><a href="#Construction-of-an-RKHS-from-a-kernel-Moore-Aronsaj" class="headerlink" title="Construction of an RKHS from a kernel: Moore-Aronsaj"></a>Construction of an RKHS from a kernel: Moore-Aronsaj</h2>
      
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